Markov functionals of an ergodic Markov process

We say that a process $(\xi(t))_{t\ge 0}$ is a Markov functional of a basic homogeneous Markov process $(X(t))_{t\ge 0}$ if the pair $(X(t),\xi(t))_{t\ge 0}$ is a Markov process. In the paper a sequence of Markov functionals $(\xi_n(t))_{t\ge 0}$ of the basic process $(X(t))_{t\ge 0}$, which is degenerate in the limit, is considered and the limit behavior of the distribution of the pair $(X(t),\xi(t))_{t\ge 0}$ is studied as $n\to\infty$.